Video taping of shows with copyrighted music during public OR private events is technically illegal. There is no difference between private and public shows.

The issue is the type of clearance obtained by the show producers. The music used have specific restrictions and in most cases the videotaping for any purposes whatsoever (including personal use) is not permitted unless the producers of the show requested a specific clearance. This applies both to public and private shows.

Does this mean that the music labels will come knocking on your door because you taped your 6 year old daughter performing in a play? Very unlikely!!

But technically the videotaping of events without a explicit clearance to do so

Yes. You need to talk to the producer and get a copy of the music release they obtained. It will likely not allow videotaping of the performance for any reason. However, you and the producer can contact the rights holder/manager and purchase the correct license - which most times is very expensive. In the future, if you get hired for something like this explain the issues to the producer and demand the obtain the correct clearance and provide you a copy of it. Sometimes you have to be named on the contract as the person authorized to videotape the performance.